Deterministic construction of arbitrary $W$ states with quadratically increasing number of two-qubit gates

Understand

We propose a quantum circuit composed of $cNOT$ gates and four single-qubit gates to generate a $W$ state of three qubits.

• This circuit was then enhanced by integrating two-qubit gates to create a $W$ state of four and five qubits.
• After a couple of enhancements, we show that an arbitrary $W$ state can be generated depending only on the degree of enhancement.
• The generalized formula for the number of two-qubit gates required is given, showing that an $n$-qubit $W$-state generation can be achieved with quadratically increasing number of two-qubit gates.